1: *> \brief \b ZGEBAL
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZGEBAL + dependencies
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11: *> [TGZ]</a>
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13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgebal.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
22: *
23: * .. Scalar Arguments ..
24: * CHARACTER JOB
25: * INTEGER IHI, ILO, INFO, LDA, N
26: * ..
27: * .. Array Arguments ..
28: * DOUBLE PRECISION SCALE( * )
29: * COMPLEX*16 A( LDA, * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> ZGEBAL balances a general complex matrix A. This involves, first,
39: *> permuting A by a similarity transformation to isolate eigenvalues
40: *> in the first 1 to ILO-1 and last IHI+1 to N elements on the
41: *> diagonal; and second, applying a diagonal similarity transformation
42: *> to rows and columns ILO to IHI to make the rows and columns as
43: *> close in norm as possible. Both steps are optional.
44: *>
45: *> Balancing may reduce the 1-norm of the matrix, and improve the
46: *> accuracy of the computed eigenvalues and/or eigenvectors.
47: *> \endverbatim
48: *
49: * Arguments:
50: * ==========
51: *
52: *> \param[in] JOB
53: *> \verbatim
54: *> JOB is CHARACTER*1
55: *> Specifies the operations to be performed on A:
56: *> = 'N': none: simply set ILO = 1, IHI = N, SCALE(I) = 1.0
57: *> for i = 1,...,N;
58: *> = 'P': permute only;
59: *> = 'S': scale only;
60: *> = 'B': both permute and scale.
61: *> \endverbatim
62: *>
63: *> \param[in] N
64: *> \verbatim
65: *> N is INTEGER
66: *> The order of the matrix A. N >= 0.
67: *> \endverbatim
68: *>
69: *> \param[in,out] A
70: *> \verbatim
71: *> A is COMPLEX*16 array, dimension (LDA,N)
72: *> On entry, the input matrix A.
73: *> On exit, A is overwritten by the balanced matrix.
74: *> If JOB = 'N', A is not referenced.
75: *> See Further Details.
76: *> \endverbatim
77: *>
78: *> \param[in] LDA
79: *> \verbatim
80: *> LDA is INTEGER
81: *> The leading dimension of the array A. LDA >= max(1,N).
82: *> \endverbatim
83: *>
84: *> \param[out] ILO
85: *> \verbatim
86: *> ILO is INTEGER
87: *> \endverbatim
88: *>
89: *> \param[out] IHI
90: *> \verbatim
91: *> IHI is INTEGER
92: *> ILO and IHI are set to INTEGER such that on exit
93: *> A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
94: *> If JOB = 'N' or 'S', ILO = 1 and IHI = N.
95: *> \endverbatim
96: *>
97: *> \param[out] SCALE
98: *> \verbatim
99: *> SCALE is DOUBLE PRECISION array, dimension (N)
100: *> Details of the permutations and scaling factors applied to
101: *> A. If P(j) is the index of the row and column interchanged
102: *> with row and column j and D(j) is the scaling factor
103: *> applied to row and column j, then
104: *> SCALE(j) = P(j) for j = 1,...,ILO-1
105: *> = D(j) for j = ILO,...,IHI
106: *> = P(j) for j = IHI+1,...,N.
107: *> The order in which the interchanges are made is N to IHI+1,
108: *> then 1 to ILO-1.
109: *> \endverbatim
110: *>
111: *> \param[out] INFO
112: *> \verbatim
113: *> INFO is INTEGER
114: *> = 0: successful exit.
115: *> < 0: if INFO = -i, the i-th argument had an illegal value.
116: *> \endverbatim
117: *
118: * Authors:
119: * ========
120: *
121: *> \author Univ. of Tennessee
122: *> \author Univ. of California Berkeley
123: *> \author Univ. of Colorado Denver
124: *> \author NAG Ltd.
125: *
126: *> \ingroup complex16GEcomputational
127: *
128: *> \par Further Details:
129: * =====================
130: *>
131: *> \verbatim
132: *>
133: *> The permutations consist of row and column interchanges which put
134: *> the matrix in the form
135: *>
136: *> ( T1 X Y )
137: *> P A P = ( 0 B Z )
138: *> ( 0 0 T2 )
139: *>
140: *> where T1 and T2 are upper triangular matrices whose eigenvalues lie
141: *> along the diagonal. The column indices ILO and IHI mark the starting
142: *> and ending columns of the submatrix B. Balancing consists of applying
143: *> a diagonal similarity transformation inv(D) * B * D to make the
144: *> 1-norms of each row of B and its corresponding column nearly equal.
145: *> The output matrix is
146: *>
147: *> ( T1 X*D Y )
148: *> ( 0 inv(D)*B*D inv(D)*Z ).
149: *> ( 0 0 T2 )
150: *>
151: *> Information about the permutations P and the diagonal matrix D is
152: *> returned in the vector SCALE.
153: *>
154: *> This subroutine is based on the EISPACK routine CBAL.
155: *>
156: *> Modified by Tzu-Yi Chen, Computer Science Division, University of
157: *> California at Berkeley, USA
158: *> \endverbatim
159: *>
160: * =====================================================================
161: SUBROUTINE ZGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
162: *
163: * -- LAPACK computational routine --
164: * -- LAPACK is a software package provided by Univ. of Tennessee, --
165: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
166: *
167: * .. Scalar Arguments ..
168: CHARACTER JOB
169: INTEGER IHI, ILO, INFO, LDA, N
170: * ..
171: * .. Array Arguments ..
172: DOUBLE PRECISION SCALE( * )
173: COMPLEX*16 A( LDA, * )
174: * ..
175: *
176: * =====================================================================
177: *
178: * .. Parameters ..
179: DOUBLE PRECISION ZERO, ONE
180: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
181: DOUBLE PRECISION SCLFAC
182: PARAMETER ( SCLFAC = 2.0D+0 )
183: DOUBLE PRECISION FACTOR
184: PARAMETER ( FACTOR = 0.95D+0 )
185: * ..
186: * .. Local Scalars ..
187: LOGICAL NOCONV
188: INTEGER I, ICA, IEXC, IRA, J, K, L, M
189: DOUBLE PRECISION C, CA, F, G, R, RA, S, SFMAX1, SFMAX2, SFMIN1,
190: $ SFMIN2
191: * ..
192: * .. External Functions ..
193: LOGICAL DISNAN, LSAME
194: INTEGER IZAMAX
195: DOUBLE PRECISION DLAMCH, DZNRM2
196: EXTERNAL DISNAN, LSAME, IZAMAX, DLAMCH, DZNRM2
197: * ..
198: * .. External Subroutines ..
199: EXTERNAL XERBLA, ZDSCAL, ZSWAP
200: * ..
201: * .. Intrinsic Functions ..
202: INTRINSIC ABS, DBLE, DIMAG, MAX, MIN
203: *
204: * Test the input parameters
205: *
206: INFO = 0
207: IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.LSAME( JOB, 'P' ) .AND.
208: $ .NOT.LSAME( JOB, 'S' ) .AND. .NOT.LSAME( JOB, 'B' ) ) THEN
209: INFO = -1
210: ELSE IF( N.LT.0 ) THEN
211: INFO = -2
212: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
213: INFO = -4
214: END IF
215: IF( INFO.NE.0 ) THEN
216: CALL XERBLA( 'ZGEBAL', -INFO )
217: RETURN
218: END IF
219: *
220: K = 1
221: L = N
222: *
223: IF( N.EQ.0 )
224: $ GO TO 210
225: *
226: IF( LSAME( JOB, 'N' ) ) THEN
227: DO 10 I = 1, N
228: SCALE( I ) = ONE
229: 10 CONTINUE
230: GO TO 210
231: END IF
232: *
233: IF( LSAME( JOB, 'S' ) )
234: $ GO TO 120
235: *
236: * Permutation to isolate eigenvalues if possible
237: *
238: GO TO 50
239: *
240: * Row and column exchange.
241: *
242: 20 CONTINUE
243: SCALE( M ) = J
244: IF( J.EQ.M )
245: $ GO TO 30
246: *
247: CALL ZSWAP( L, A( 1, J ), 1, A( 1, M ), 1 )
248: CALL ZSWAP( N-K+1, A( J, K ), LDA, A( M, K ), LDA )
249: *
250: 30 CONTINUE
251: GO TO ( 40, 80 )IEXC
252: *
253: * Search for rows isolating an eigenvalue and push them down.
254: *
255: 40 CONTINUE
256: IF( L.EQ.1 )
257: $ GO TO 210
258: L = L - 1
259: *
260: 50 CONTINUE
261: DO 70 J = L, 1, -1
262: *
263: DO 60 I = 1, L
264: IF( I.EQ.J )
265: $ GO TO 60
266: IF( DBLE( A( J, I ) ).NE.ZERO .OR. DIMAG( A( J, I ) ).NE.
267: $ ZERO )GO TO 70
268: 60 CONTINUE
269: *
270: M = L
271: IEXC = 1
272: GO TO 20
273: 70 CONTINUE
274: *
275: GO TO 90
276: *
277: * Search for columns isolating an eigenvalue and push them left.
278: *
279: 80 CONTINUE
280: K = K + 1
281: *
282: 90 CONTINUE
283: DO 110 J = K, L
284: *
285: DO 100 I = K, L
286: IF( I.EQ.J )
287: $ GO TO 100
288: IF( DBLE( A( I, J ) ).NE.ZERO .OR. DIMAG( A( I, J ) ).NE.
289: $ ZERO )GO TO 110
290: 100 CONTINUE
291: *
292: M = K
293: IEXC = 2
294: GO TO 20
295: 110 CONTINUE
296: *
297: 120 CONTINUE
298: DO 130 I = K, L
299: SCALE( I ) = ONE
300: 130 CONTINUE
301: *
302: IF( LSAME( JOB, 'P' ) )
303: $ GO TO 210
304: *
305: * Balance the submatrix in rows K to L.
306: *
307: * Iterative loop for norm reduction
308: *
309: SFMIN1 = DLAMCH( 'S' ) / DLAMCH( 'P' )
310: SFMAX1 = ONE / SFMIN1
311: SFMIN2 = SFMIN1*SCLFAC
312: SFMAX2 = ONE / SFMIN2
313: 140 CONTINUE
314: NOCONV = .FALSE.
315: *
316: DO 200 I = K, L
317: *
318: C = DZNRM2( L-K+1, A( K, I ), 1 )
319: R = DZNRM2( L-K+1, A( I, K ), LDA )
320: ICA = IZAMAX( L, A( 1, I ), 1 )
321: CA = ABS( A( ICA, I ) )
322: IRA = IZAMAX( N-K+1, A( I, K ), LDA )
323: RA = ABS( A( I, IRA+K-1 ) )
324: *
325: * Guard against zero C or R due to underflow.
326: *
327: IF( C.EQ.ZERO .OR. R.EQ.ZERO )
328: $ GO TO 200
329: G = R / SCLFAC
330: F = ONE
331: S = C + R
332: 160 CONTINUE
333: IF( C.GE.G .OR. MAX( F, C, CA ).GE.SFMAX2 .OR.
334: $ MIN( R, G, RA ).LE.SFMIN2 )GO TO 170
335: IF( DISNAN( C+F+CA+R+G+RA ) ) THEN
336: *
337: * Exit if NaN to avoid infinite loop
338: *
339: INFO = -3
340: CALL XERBLA( 'ZGEBAL', -INFO )
341: RETURN
342: END IF
343: F = F*SCLFAC
344: C = C*SCLFAC
345: CA = CA*SCLFAC
346: R = R / SCLFAC
347: G = G / SCLFAC
348: RA = RA / SCLFAC
349: GO TO 160
350: *
351: 170 CONTINUE
352: G = C / SCLFAC
353: 180 CONTINUE
354: IF( G.LT.R .OR. MAX( R, RA ).GE.SFMAX2 .OR.
355: $ MIN( F, C, G, CA ).LE.SFMIN2 )GO TO 190
356: F = F / SCLFAC
357: C = C / SCLFAC
358: G = G / SCLFAC
359: CA = CA / SCLFAC
360: R = R*SCLFAC
361: RA = RA*SCLFAC
362: GO TO 180
363: *
364: * Now balance.
365: *
366: 190 CONTINUE
367: IF( ( C+R ).GE.FACTOR*S )
368: $ GO TO 200
369: IF( F.LT.ONE .AND. SCALE( I ).LT.ONE ) THEN
370: IF( F*SCALE( I ).LE.SFMIN1 )
371: $ GO TO 200
372: END IF
373: IF( F.GT.ONE .AND. SCALE( I ).GT.ONE ) THEN
374: IF( SCALE( I ).GE.SFMAX1 / F )
375: $ GO TO 200
376: END IF
377: G = ONE / F
378: SCALE( I ) = SCALE( I )*F
379: NOCONV = .TRUE.
380: *
381: CALL ZDSCAL( N-K+1, G, A( I, K ), LDA )
382: CALL ZDSCAL( L, F, A( 1, I ), 1 )
383: *
384: 200 CONTINUE
385: *
386: IF( NOCONV )
387: $ GO TO 140
388: *
389: 210 CONTINUE
390: ILO = K
391: IHI = L
392: *
393: RETURN
394: *
395: * End of ZGEBAL
396: *
397: END
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